I am trying to factor$x^5-5x^3+4x$ so that I can find the roots. I know from the answers section that the roots are where $x = 0, 1, -1, 2$ and $-2$.
I'm stuck, here's as far as I got:
$$
x^5-5x^3+4x =
x(x^4-5x^2+4)
$$
Let $u = x^2$ and just focus on the term on the right (drop the first $x$ for now):
$$x^4-5x^2+4 = u^2-5u+4x.$$
Master term is $a \times c = 1 \times 4 = 4$.
Seeking a pair of numbers that sum to the middle term $-5$ and whose product is $4$:
\begin{align}
1 \times -4 &= -4, &\text{sum} &= -3 \\
4 \times -1 &= -4, &\text{sum} &= 3 \\
2 \times -2 &= -4, &\text{sum} &= 0
\end{align}
???
I'm not sure how to proceed since I cannot find a pair of numbers that satisfy the condition.
Have I gone wrong somewhere? How can I factor $u^2-5u+4x$?
Best Answer
$$x^5-5x^3+4x=x(x^4-5x^2+4)$$$$=x(x^4-4x^2-x^2+4)$$$$=x(x^2(x^2-4)-(x^2-4))$$$$=x((x^2-4)(x^2-1))$$ $$=x(x+2)(x-2)(x+1)(x-1)$$